Maximum even factors of graphs

A spanning subgraph F of a graph G is called an even factor of G if each vertex of F has even degree at least 2 in F. Kouider and Favaron proved that if a graph G has an even factor, then it has an even factor F   with |E(F)|≥916(|E(G)|+1). In this paper we improve the coefficient 916 to 47, which is best possible. Furthermore, we characterize all the extremal graphs, showing that if |E(H)|≤47(|E(G)|+1) for every even factor H of G, then G belongs to a specified class of graphs.